# Stochastic gene expression conditioned on large deviations

## TL;DR

This paper applies advanced statistical mechanics and queueing theory to analyze large deviations in stochastic gene expression models, providing exact results for rare event probabilities and system fluctuations.

## Contribution

It introduces a framework combining non-equilibrium statistical mechanics and queueing theory to analyze large deviations in gene expression modeled as BMAPs, revealing renormalized parameters for conditioned processes.

## Key findings

- Exact analytical results for large deviations in gene expression models
- Representation of conditioned processes with renormalized parameters
- Identification of parameters leading to dynamical phase transitions

## Abstract

The intrinsic stochasticity of gene expression can give rise to large fluctuations and rare events that drive phenotypic variation in a population of genetically identical cells. Characterizing the fluctuations that give rise to such rare events motivates the analysis of large deviations in stochastic models of gene expression. Recent developments in non-equilibrium statistical mechanics have led to a framework for analyzing Markovian processes conditioned on rare events and for representing such processes by conditioning-free driven Markovian processes. We use this framework, in combination with approaches based on queueing theory, to analyze a general class of stochastic models of gene expression. Modeling gene expression as a Batch Markovian Arrival Process (BMAP), we derive exact analytical results quantifying large deviations of time-integrated random variables such as promoter activity fluctuations. We find that the conditioning-free driven process can also be represented by a BMAP that has the same form as the original process, but with renormalized parameters. The results obtained can be used to quantify the likelihood of large deviations, to characterize system fluctuations conditional on rare events and to identify combinations of model parameters that can give rise to dynamical phase transitions in system dynamics.